Equations are helpful for communicating how quantities are changing. We can write equations from descriptions or from graphs.

Sometimes, the information on how a quantity is changing is given in a graph instead of in words. We can find an equation for an exponential function using two points on its graph, just as we’ve done in the past with linear functions. Let’s say we want to find a function of the form , whose graph contains and .

Because , the -intercept of the graph is . In this example, the intercept is given as , so we know that .

Using the second given point, , we know . This means that . Solving this equation we have:

To determine the exact value of , let’s use the properties of exponents. Since is positive, we can show that because

We can now write an equation defining : .